geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, circular symmetry is a type of

continuous symmetry In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant u ...

for a planar object that can be rotated by any arbitrary

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

and map onto itself. Rotational circular symmetry is isomorphic with the

circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \ ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

, or the special orthogonal group SO(2), and

unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...

U(1). Reflective circular symmetry is isomorphic with the

orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...

O(2).

Two dimensions

A 2-dimensional object with circular symmetry would consist of

concentric circle In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...

s and

annular Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * '' Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus co ...

domains. Rotational circular symmetry has all

cyclic symmetry In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bin ...

, Z_''n'' as subgroup symmetries. Reflective circular symmetry has all

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...

, Dih_''n'' as subgroup symmetries.

Three dimensions

In 3-dimensions, a

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

solid of revolution In geometry, a solid of revolution is a solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution'') that lies on the same plane. The surface created by this revolution and which bounds the solid is the ...

has circular symmetry around an axis, also called cylindrical symmetry or axial symmetry. An example is a right circular

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

. Circular symmetry in 3 dimensions has all

pyramidal symmetry In three dimensional geometry, there are four infinite series of point groups in three dimensions (''n''≥1) with ''n''-fold rotational or reflectional symmetry about one axis (by an angle of 360°/''n'') that does not change the object. They are ...

, C_''n''v as subgroups. A double-cone,

bicone In geometry, a bicone or dicone (from la, bi-, and Greek: ''di-'', both meaning "two") is the three-dimensional surface of revolution of a rhombus around one of its axes of symmetry. Equivalently, a bicone is the surface created by joining ...

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...

and

spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...

have circular symmetry, and in addition have a

bilateral symmetry Symmetry in biology refers to the symmetry observed in organisms, including plants, animals, fungi, and bacteria. External symmetry can be easily seen by just looking at an organism. For example, take the face of a human being which has a pla ...

perpendular to the axis of system (or half cylindrical symmetry). These reflective circular symmetries have all discrete prismatic symmetries, D_''n''h as subgroups.

Four dimensions

In four dimensions, an object can have circular symmetry, on two orthogonal axis planes, or duocylindrical symmetry. For example, the

duocylinder The duocylinder, also called the double cylinder or the bidisc, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disks of respective radii ''r''1 and ''r''2: :D = \left\ It is analogous ...

and

Clifford torus In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...

have circular symmetry in two orthogonal axes. A

spherinder In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3- ball (or solid 2-sphere) of radius ''r''1 and a line segment of length 2''r''2: :D = \ Lik ...

has spherical symmetry in one 3-space, and circular symmetry in the orthogonal direction.

Spherical symmetry

An analogous 3-dimensional equivalent term is spherical symmetry. Rotational spherical symmetry is isomorphic with the

rotation group SO(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...

, and can be parametrized by the Davenport chained rotations pitch, yaw, and roll. Rotational spherical symmetry has all the discrete chiral 3D

point group In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every p ...

s as subgroups. Reflectional spherical symmetry is isomorphic with the

O(3) and has the 3-dimensional discrete point groups as subgroups. A

scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...

has spherical symmetry if it depends on the distance to the origin only, such as the

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...

of a

central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...

. A vector field has spherical symmetry if it is in radially inward or outward direction with a magnitude and orientation (inward/outward) depending on the distance to the origin only, such as a central force.

References

* * *{{springer, title=Orthogonal group, id=p/o070300 Symmetry Rotation

Two dimensions

Three dimensions

Four dimensions

Spherical symmetry

See also

References